lesson 1 - biuu.cs.biu.ac.il/~rosenfa5/ai/lessons/allin1.pdf · 8 ai – alternative definitions...

1

Artificial Intelligence

Lesson 1

2

About

• Lecturer: Prof. Sarit Kraus

• TA: Ariel Rosenfeld

• (almost) All you need can be found on the

course website:

– http://u.cs.biu.ac.il/~rosenfa5/

3

Course Requirements 1

• The grade is comprised of 70% exam and 30% exercises.

• 3 programming exercises will be given. Work individually.

• All the exercises are counted for the final grade, but you

can pass the course without submitting them if your final

grade (composed from the exam and exercises grades) is

above the required threshold. The exercises are equally

counted.

• Exercises will be written in C++ or JAVA only. They

should compile and run on planet machine, and will be

submitted via “submit”. Be precise!

4

Course Requirements 2 • Exercises are not hard, but work is required. Plan your

time ahead!

• When sending me mail please include the course number

(89-570) in the header, to pass the automatic spam filter.

• You (probably) will be required to participate in AI

experiments.

• See other general rules in: http://u.cs.biu.ac.il/~haimga/Teaching/AI/assignments/general-rules.pdf

5

Course Schedule

• Lesson 1:

– Introduction

– Transferring a general problem to a graph search problem.

• Lesson 2

– Uninformed Search (BFS, DFS etc.).

• Lesson 3

– Informed Search (A*,Best-First-Search etc.).

6

Course Schedule – Cont.

• Lesson 4

– Local Search (Hill Climbing, Genetic algorithms etc.).

• Lesson 5

– “Search algorithms” chapter summery.

• Lesson 6-7

– Game-Trees: Min-Max & Alpha-Beta algorithms.

7

Course Schedule – Cont.

• Lesson 8-9

– Planning: STRIPS algorithm

• Lesson 10-11-12

– Learning: Decision-Trees, Neural Network,

Naïve Bayes, Bayesian Networks and more.

• Lesson 13

– Questions and exercise.

8

AI – Alternative Definitions

• Elaine Rich and Kevin Knight: AI is the study of how to make computers do things at which, at the moment, people are better.

• Stuart Russell and Peter Norvig: [AI] has to do with smart programs, so let's get on and write some.

• Claudson Bornstein: AI is the science of common sense.

• Douglas Baker: AI is the attempt to make computers do what people think computers cannot do.

• Astro Teller: AI is the attempt to make computers do what they do in the movies.

9

AI Domains

• Games – chess, checkers, tile puzzle.

• Expert systems

• Speech recognition and Natural language

processing, Computer vision, Robotics.

10

AI & Search

• "The two most fundamental concerns of AI researchers are knowledge representation and search”

• “knowledge representation … addresses the problem of capturing in a language…suitable for computer manipulation”

• “Search is a problem-solving technique that systematically explores a space of problem states”.Luger, G.F. Artificial Intelligence: Structures and Strategies for

Complex Problem Solving

11

Solving Problems with Search

Algorithms

• Input: a problem P.

• Preprocessing:

– Define states and a state space

– Define Operators

– Define a start state and goal set of states.

• Processing:

– Activate a Search algorithm to find a path form start to one of the goal states.

12

Example - Missionaries & Cannibals

• State space – [M,C,B]

• Initial State – [3,3,1]

• Goal State – [0,0,0]

• Operators – adding or subtracting the vectors [1,0,1], [2,0,1], [0,1,1], [0,2,1] or [1,1,1]

• Path – moves from [3,3,1] to [0,0,0]

• Path Cost – river trips

13

Breadth-First-Search Pseudo code

• Intuition: Treating the graph as a tree and scanning top-

down.

• Algorithm:

BFS(Graph graph, Node start, Vector Goals)

1. L make_queue(start)

2. While L not empty loop

1. n L.remove_front()

2. If goal (n) return true

3. S successors (n)

4. L.insert(S)

3. Return false

14

Breadth-First-Search Attributes

• Completeness – yes

• Optimality – yes, if graph is un-

weighted.

• Time Complexity:

• Memory Complexity:

– Where b is branching factor and d is the

solution depth

• See water tanks example.

( , )b d

)( 1dbO

)()...1( 112 dd bObbbbO

15

Artificial Intelligence

Lesson 2

16

Uninformed Search

• Uninformed search methods use only information available in the problem definition.

– Breadth First Search (BFS)

– Depth First Search (DFS)

– Iterative DFS (IDA)

– Bi-directional search

– Uniform Cost Search (a.k.a. Dijkstra alg.)

17

Depth-First-Search Pseudo code

DFS(Graph graph, Node start, Vector Goals)

1. L make_stack(start)

2. While L not empty loop

2.1 n L.remove_front()

2.2 If goal (n) return true

2.3 S successors (n)

2.4 L.insert(S)

3. Return false

18

Depth-First-Search Attributes

• Completeness – No. Infinite loops or

Infinite depth can occur.

• Optimality – No.

• Time Complexity:

• Memory Complexity:

– Where b is branching factor and m is the

maximum depth of search tree

• See water tanks example

4

1

2

3

5

( )mO b

( )O bm

19

Limited DFS Attributes

• Completeness – Yes, if d≤l

• Optimality – No.

• Time Complexity:

– If d<l, it is larger than in BFS

• Memory Complexity:

– Where b is branching factor and l is the

depth limit.

( )lO b

( )O bl

20

Depth-First Iterative-Deepening

The numbers represent the order generated by DFID

0

1,3,

9 2,6,16

c 4,10 5,13 c 7,17 8,20

11 12 21 22 c 14 15 18 19

21

Iterative-Deepening Attributes

• Completeness – Yes

• Optimality – yes, if graph is un-weighted.

• Time Complexity:

• Memory Complexity: ( )O db

– Where b is branching factor and d is the maximum

depth of search tree

)())1(...)1()(( 2 dd bObbdbdO

22

State Redundancies

• Closed list - a hash table which holds the

visited nodes.

• For example BFS:

Closed List

Open List (Frontier)

23

Bi-directional Search

• Search both from initial state to goal state.

• Operators must be symmetric.

S G

24

Bi-directional Search Attributes

• Completeness – Yes, if both directions use BFS

• Optimality – yes, if graph is un-weighted and both

directions use BFS.

• Time and memory Complexity:

• Pros.

– Cuts the search tree by half (at least theoretically).

• Cons.

– Frontiers must be constantly compared.

)( 2/dbO

25

Minimum cost path

• General minimum cost path-search

problem:

– Find shortest path from start state to one of the

goal states in a weighted graph.

– Path cost function is g(n): sum of weights from

start state to goal.

26

Uniform Cost Search

• Also known as Dijkstra’s algorithm.

• Expand the node with the minimum path

cost first.

• Implementation: priority queue.

27

Uniform Cost Search Attributes

• Completeness: yes, for positive weights

• Optimality: yes

• Time & Memory complexity:

– Where b is branching factor, c is the optimal solution cost

and e is the minimum edge cost

)(/ec

bO

28

Example of Uniform Cost Search • Assume an example tree with different edge costs, represented by

numbers next to the edges.

Notations for this example:

generated node

expanded node

a

c

b c

2 1

1

2

c c f

1

2

g d e

29

Example of Uniform Cost Search

Closed list:

Open list:

a 2 1

0

a

30

Example of Uniform Cost Search

Closed list:

Open list:

a

b c

2 1

1

2

1

2

2 1

b c

a

31

Example of Uniform Cost Search

Closed list:

Open list:

a

c

b c

2 1

1

2

c

1

2

d e

2 2 3

b d e

a c

32

Example of Uniform Cost Search

Closed list:

Open list:

a

c

b c

2 1

1

2

c c f

1

2

g d e

a c b

2 3 3 4

d e f g

33

Example of Uniform Cost Search

Closed list:

Open list:

a

c

b c

2 1

1

2

c c f

1

2

g d e

3 3 4

e f g

a c b d

34

Example of Uniform Cost Search

Closed list:

Open list:

a

c

b c

2 1

1

2

c c f

1

2

g d e

3 4

f g

a c b d e

35

Example of Uniform Cost Search

Closed list:

Open list:

a

c

b c

2 1

1

2

c c f

1

2

g d e

4

g

a c b d e f

36

Example of Uniform Cost Search

Closed list:

Open list:

a

c

b c

2 1

1

2

c c f

1

2

g d e

a c b d e f g

37

Informed Search

• Incorporate additional measure of a

potential of a specific state to reach the

goal.

• A potential of a state to reach a goal is

measured through a heuristic function h(n).

• An evaluation function is denoted f(n).

38

Best First Search Algorithms

• Principle: Expand node n with the best evaluation function value f(n).

• Implement via a priority queue

• Algorithms differ with definition of f :

– Greedy Search:

– A*:

– IDA*: iterative deepening version of A*

– Etc’

( ) ( )f n h n

( ) ( ) ( )f n g n h n

39

Exercise

• Q: Does a Uniform-Cost search be considered as a Best-First algorithm?

• A: Yes. It can be considered as a Best-First algorithm with evaluation function f(n)=g(n).

• Q: In what scenarios IDS outperforms DFS?, BFS?

• A: – IDS outperforms DFS when the search tree is a lot

deeper than the solution depth.

– IDS outperforms BFS when BFS run out of memory.

40

Exercise – Cont.

• Q: Why do we need a closed list?

• A: Generally a closed list has two main functionalities:

– Prevent re-exploring of nodes.

– Hold solution path from start to goal (DFS based algorithms have it anyway).

• Q: Does Breadth-FS find optimal path length in general?

• A: No, unless the search graph is un-weighted.

• Q: Will IDS always find the same solution as BFS given that the nodes expansion order is deterministic?

• A: Yes. Each iteration of IDS explores new nodes the same order a BFS does.

41

Artificial Intelligence

Lesson 3

42

Informed Search

• Incorporate additional measure of a

potential of a specific state to reach the

goal.

• A potential of a state to reach a goal is

measured through a heuristic function h(n),

thus always h(goal) = 0.

• An evaluation function is denoted f(n).

43

Best First Search Algorithms

• Principle: Expand node n with the best evaluation function value f(n).

• Implement via a priority queue

• Algorithms differ with definition of f :

– Greedy Search:

– A*:

– IDA*: iterative deepening version of A*

– Etc’.

( ) ( )f n h n

( ) ( ) ( )f n g n h n

44

Properties of Heuristic functions

• The 2 most important properties:

– relatively cheap to compute

– relatively accurate estimator of the cost to reach a goal.

Usually a “good” heuristic is if ½ opt(n)<h(n)≤opt(n)

• Examples:

– Navigating in a network of roads from one location to

another. Heuristic function: Airline distance.

– Sliding-tile puzzles. Heuristic function: Manhattan

distance - number of horizontal and vertical grid units each

tile is displaced from its goal position

45

Heuristic Function h(n)

• Admissible/Underestimate: h(n) never

overestimate the actual cost from n to goal

• Consistent/monotonic (desirable) :

h(m)-h(n) ≤w(n,m) where m is parent of n. This

ensures f(n) ≥f(m).

46

Best-FS Algorithm Pseudo code

1. Start with open = [initial-state].

2. While open is not empty do

1. Pick the best node on open.

2. If it is the goal node then return with success.

Otherwise find its successors.

3. Assign the successor nodes a score using the

evaluation function and add the scored nodes

to open

47

General Framework using Closed-

list (Graph-Search)

GraphSearch(Graph graph, Node start, Vector goals)

1. O make_data_structure(start) // open list

2. Cmake_hash_table // closed list

3. While O not empty loop

1. n O.remove_front()

2. If goal (n) return n

3. If n is found on C continue

4. //otherwise

5. O successors (n)

6. Cn

4. Return null //no goal found

48

Greedy Search Attributes

• Completeness: No. Inaccurate heuristics can

cause loops (unless using a closed list), or

entering an infinite path

• Optimality: No. Inaccurate heuristics can

lead to a non optimal solution.

• Time & Memory complexity:

( )mO b

s

g

a b

1 3

1

2

h=2 h=1

A* Algorithm

• Combines greedy h(n) and uniform cost g(n) approaches.

• Evaluation function: f(n)=g(n)+h(n)

49

50

A* Pseudo code A-Star(Graph graph, Node start, Node goal, HeuristicFunction h)

1. O make_priority_queue(startNode) // open list

2. Cmake_hash_table // closed list

3. While O not empty loop

1. n O.remove_front() //O is sorted by f(n)=g(n)+h(n) values

2. If goal (n) return n

3. If n is found on C continue

4. //otherwise

5. S successors (n)

6. For each node s in S

1. Set s.g=n.g+w(n,s)

2. Set s.parent=n //for path extraction

3. Set s.h=h(s) //to calculate f

4. Os

7. Cn

4. Return null //no goal found

51

A* Algorithm (1) • Completeness:

– In a finite graph: Yes

– In an infinite graph: if all edge costs are finite and have a minimum positive value, and all heuristic values are finite and non-negative.

• Optimality: – In tree-search: if h(n) is admissible

– In graph-search: if it is also consistent

52

A* Algorithm (2)

• optimally efficient: A* expands the

minimal number of nodes possible with any

given (consistent) heuristic.

• Time and space complexity:

– Worst case: Cost function f(n) = g(n)

– Best case: Cost function f(n) = g(n) + h*(n)

)( /ecbO

)(bdO

53

A* Application Example

• Game: Tales of Trolls

and Treasures

• Yellow dots are nodes

in the search graph.

54

IDA* Algorithm

• Each iteration is a depth-first search that keeps track of the cost evaluation f = g + h of each node generated.

• The cost threshold is initialized to the heuristic of the initial state.

• If a node is generated whose cost exceeds the threshold for that iteration, its path is cut off.

55

IDA* Pseudo code

• IDAStar-Main (Node root)

1. Set bound = f(root);

2. WHILE (bound<infinity) 1. Set bound= IDAStar(root, bound)

• IDAStar(node n, Double bound)

1. if n is a goal, Exit algorithm and return goal

2. if n has no children, return infinity

3. fn = infinity

4. for each child c of n, Set f=f(c ) 1. IF (f<= bound) fn=min(fn, IDAStar(c,bound))

2. Else fn=min(fn,f)

5. Return fn

56

IDA* Attributes

• The cost threshold increases in each iteration to the total cost of the lowest-cost node that was pruned during the previous iteration.

• The algorithm terminates when a goal state is reached whose total cost does not exceed the current threshold.

• Completeness and Optimality: Like A*

• Space complexity:

• Time complexity*: )( /ecbO

)(cO

57

Duplicate Pruning

• Do not enter the father of the current state

– With or without using closed-list

• Using a closed-list, check the closed list before

entering new nodes to the open list

– Note: in A*, h has to be consistent!

– Do not remove the original check

• Using a stack, check the current branch and

stack status before entering new nodes

58

Exercise

• Q: What are the

advantages of IDA*

over:

– A*?

– DFS (no closed list)?

– Uniform-Cost (closed

list)?

Optimality Space Informed

pruning

Endless

branch

Alg.

Adv.

V A*

V V V DFS

V V UC

59

Exercise – Cont. • Q: When IDA* is not preferable?

• A:

– A space graph with dense node duplications

– When all the node costs are different, if the asymptotic complexity of A* is O(N) - IDA*‘s complexity can get in the worst case to O(N2).

• Q: What algorithm we’ll get if we implement Greedy search on a uniform cost graph using

– h(n)= g(n) ?

– h(n)= -g(n) ?

• A:

– h(n)= g(n) BFS

– h(n)= -g(n) DFS

60

Exercise – True/False.

Sentence

DFS is not optimal

Forward Search is always more

preferable than Backwards Search

ID alg. is always equal or slower

than BFS (assuming nodes

expansion order is deterministic)

IDS alg. is the exact

implementation of BFS

db

True/False

True, see DFS slides for example

False, For example if there are

more start nodes than goal nodes,

or it is more natural to go

backwards (expert systems).

True. The last iteration expands

nodes as BFS.

False. its space complexity is bd

instead of .

61

Artificial Intelligence

Lesson 4

אלגוריתמים המבצעים שיפור איטרטיבי

עבור בעיות בהן המטרה לא ידועה.

להרוויח כמה שיותר כסף - :דוגמאות.

.לארוז בכמה שפחות נפח -

.לשבץ עם כמה שפחות קונפליקטים -

יודעים רק איך להשוות בין שני מצבים ולומר מי מהם

.יותר טוב

ומנסים לעשות שינויים מקומיים כדי , מגרילים פתרון

.לשפר אותו

62

63

Local Search • Local improvement, no paths

• Look around at states in the local neighborhood and choose the one with the best value

• Pros: - Quick (usually linear)

– Sometimes enough

– Linear space complexity

– can often find reasonable solutions in large or infinite (continuous) state spaces for which systematic algorithms are unsuitable.

– Suitable for optimization problems: Math problems for finding optimal value for functions under specific constrains.

• Cons:

– Not optimal: Travelling Sale Person problem: Find the shortest path s.t every city will be visited only once.

– Can stuck on local maximum, plateau.

64

Local Search – Cont.

• In order to avoid local

maximum and

plateaus we permit

moves to states with

lower values in

probability p.

• The different

algorithms differ in p.

p Algorithm

p=0 Hill

Climbing,GSAT

p=1 Random Walk

p=c (domain

specific)

Mixed Walk,

Mixed GSAT

p=acceptor(dh,

T)

Simulated

Annealing

65

Hill Climbing

states

f-value

f-value = evaluation(state)

while f-value(state) <= f-value(next-best(state))

state := next-best(state)

Hill Climbing • Always choose the next best successor

• Stop when no improvement is possible

• The problems:

– Stops in local maximum

– If the best neighbor is equal to the node, it chooses

the neighbor

– If there are some equals neighbors, choose one

randomly

– Can stuck with no progress because of all above

66

In order to avoid plateaus and

local maximum:

- Sideways move: go to sons in which their value

equal to mine

- Stochastic hill climbing: Choose the node with

the highest grade (how much its solution is

good)

- Random-restart algorithm

67

Random Restart Hill Climbing

. hill climbingבחר בנקודה רנדומאלית והרץ את 1.

אם הפתרון שמצאת טוב יותר מהפתרון הטוב 2. .שמור אותו –ביותר שנמצא עד כה

.1-חזור ל3.

.איטרציותלאחר מספר קבוע של –? מתי נסיים

שבהן איטרציותלאחר מספר קבוע של – לא נמצא שיפור לפתרון הטוב ביותר שנמצא עד

.כה

68

69 Ram Meshulam 2004 69

Random Restart Hill Climbing

f-value = evaluation(state)

Simulated Annealing

נאפשר ירידה ממצב השיא אליו , במקום להתחיל בכל פעם מחדש .הגענו

התהליך דומה לטיפוס הרים אבל בכל שלב בוחרים צעד אקראי .

אם הצעד משפר את ערך- f נבצע אותו.

נבצע אותו בהסתברות מסוימת, אחרת.

כל עוד לא מוצאים חזקתיתפונקצית ההסתברות יורדת בצורה

. פתרון

70

71

Simulated Annealing • Permits moves to states with lower values

• Gradually decreases the frequency of such moves and their size.

• Analogue to physical process of freezing liquid.

• Schedule()

– Returns the current temperature

– Depends on start temperature and round number

• Acceptor()

– Returns the probability of choosing “bad” node.

– Depends on h(n)-h(n_son) and current temperature.

72

Simulated Annealing – Pseudo code

• Simulated Annealing(start node s, Temperature t, ) 1. Set startTemp=t //for schedule function

2. Set h= h(s)

3. Set round=0

4. while terminal condition not true 1. Set s_new = choose random son of s

2. Set h_new = h(s_new)

3. if (h_new < h) or (random() < acceptor(h_new-h,t)) 1. Set s=s_new

2. Set h=h_new

3. Set t=schedule(startTemp, round)

4. Set round=round+1

73

Simulated Annealing – Pseudo code

Cont.

• Acceptor func: Decides

whether to go to a bad

node or not…example:

• Schedule func: Decrease

the temp following the

rounds. example:

roundc startTemp0<c<1

tc

h

e

10 c

74

GSAT

• Greedy local search procedure for satisfying

logic formulas in a conjunctive normal form

(CNF).

• An implementation of Hill Climbing for the

CNF domain.

• Note: SAT is NP-Complete problem.

GSAT • Searcher:

• states: variable assignments

• actions: flip a variable's assignment

• score: the number of unsatisfied clauses

• Start with a random assignment.

• While not sat...

• Flip the value assigned to the variable that yields greatest

number of satisfied clauses.

• Repeat #flips.

• Repeat with new random assignment #trials.

75

76

GSAT – Pseudo code

• GSAT(clauses C,Integer tries, Integer flips)

1. for i=1 to tries

1. Set T=a randomly generated truth assignment

2. for j= 1 to flips

1. if T satisfies C then return T

2. FLIP any variables in T that results in the greatest decrease

in the number of unsatisfied clauses

3. Save the currently best T

77

Genetic Algorithm

• Inspired by Darwin's theory of evolution:

survival of the fittest.

• Begins with a set of solutions

“chromosomes” called population.

• Best solutions from generation n are taken

and used to form a generation n+1 applying

crossover and mutation operators.

78

Genetic Algorithm Pseudo code

• choose initial population

• evaluate each individual's fitness

• repeat until terminating condition

– select individuals to reproduce //better fitness better

//chance to be selected

– mate pairs at random

– in crossover_prob. apply crossover operator

– in mutation_prob. apply mutation operator

– evaluate each individual's fitness

79

Exercise

• Q: Is there a danger of Local maximum in GA? How does the algorithm tries to avoid it?

• A: The mutation operator, which inserts randomization to the algorithm.

• Q: If start temperature very close to 0 in SA – how will the algorithm behave?

– What problem will it cause?

– How partially can we solve it?

• A: – Greedy Search with no Closed list.

– It will stuck on the first local max.

– Random-restart.

80

Exercise – Cont.

• Q: Solve the Traveling Salesman Problem using:

– Simulated annealing (SA)

– Genetic Algorithm (GA).

• A:

– For both algorithms a state is a vector which represents

the order in which the salesman travels.

– State value/fitness is the distance the agent traveled.

– State expand/mutation is to swap order of two cities in

path.

81

Exercise – Cont.

• GA:

– crossover: “greedy crossover” [greffenstette,1985]:

– GreedyCrossover(vector v1, vector v2)

1. Set vector res=v1[0] //v1 and v2 are chosen randomly

2. Repeat until |res|=number of cities

1. Select the closest city to res[i] from v1[i+1],v2[i+1] which is not already in res.

2. If not possible select randomly a city which is not in res.

82

Artificial Intelligence

Lesson 5

83

Search Algorithms Hierarchy

Local

GSAT

Hill Climbing

Random

Walk

Mixed Walk

Mixed GSAT

Simulated Annealing

Global

Informed

A*

IDA* Greedy

Uninformed

DFS

IDS

BFS

Uniform Cost

84

Exercise

• What are the different

data structures used to

implement the open

list in BFS,DFS,Best-

FS:

Queue BFS

Stack DFS

Priority

Queue

Best-FS (Greedy,A*,Unifo

rm-Cost Alg).

85

Exercise – Cont.

• If there is no solution A* will explore the

whole graph

• An admissible heuristic function h(n) will

always return smaller values than the real

distance to the goal

• h,h’ admissible A* will expand the same

number of nodes with both func.

[yes]

[no. h(n)<=h*(n) ]

[no]

86

Artificial Intelligence

Lesson 6

(From Russell & Norvig)

87

Games- Outline

• Optimal decisions

• α-β pruning

• Imperfect, real-time decisions

88

Games vs. search problems

• "Unpredictable" opponent specifying a

move for every possible opponent reply

• Time limits unlikely to find goal, must

approximate

89

Game tree (2-player,

deterministic, turns)

90

Minimax

• Perfect play for deterministic games

• Idea: choose move to position with highest minimax value

= best achievable payoff against best play

• E.g., 2-ply game:

91

Minimax algorithm

92

Properties of minimax

• Complete? (=will not run forever) Yes (if tree is finite)

• Optimal? (=will find the optimal response) Yes (against an

optimal opponent)

• Time complexity? O(bm)

• Space complexity? O(bm) (depth-first exploration), O(bm)

for saving the optimal response

• For chess, b ≈ 35, m ≈100 for "reasonable" games

exact solution completely infeasible

•

93

α-β pruning example

94

α-β pruning example

95

α-β pruning example

96

α-β pruning example

97

α-β pruning example

98

Properties of α-β

• Pruning does not affect final result

• Good move ordering improves effectiveness of pruning

• With "perfect ordering“ on binary tree, time complexity = O(bm/2)

doubles depth of search

• A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)

99

Why is it called α-β?

• α is the value of the best (i.e., highest-value) choice found so far at any choice point along the path for max

• If v is worse than α, max will avoid it

prune that branch

• Define β similarly for min

100

The α-β algorithm

101

The α-β algorithm

102

Resource limits Suppose we have 100 secs, explore 104 nodes/sec

106 nodes per move

Standard approach:

• cutoff test: e.g., depth limit

(perhaps add quiescence search: Additional “grade” for each

node) .מצב בו שני צדדים במשחק בעיצומו של החלפת כלים –בהקשר של משחקים . מצב חוסר שקט

יש סיכוי גבוה כי יחזיר ערך שגוי מאחר ותיתכן , אם אלגוריתם החיפוש יסיים לחפש בשלב כזה

. המשכת החלפת כלים נוספת

.פתרון הבעיה הוא להמשיך להעמיק בענף העץ עד שמגיעים למצב שקט שבו אין החלפת כלים

• evaluation function: estimated desirability of position

•

103

Evaluation functions

• For chess, typically linear weighted sum of features

Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s)

• e.g., w1 = 9 with

f1(s) = (number of white queens) – (number of black

queens), etc.

104

Cutting off search

MinimaxCutoff is identical to MinimaxValue except 1. "Terminal ?“ is replaced by “Cutoff?”

2. Utility is replaced by Eval

Does it work in practice?

bm = 106, b=35 m=4

4-ply lookahead is a hopeless chess player!

– 4-ply ≈ human novice

– 8-ply ≈ typical PC, human master

– 12-ply ≈ Deep Blue, Kasparov

105

Deterministic games in practice • Checkers: Chinook ended 40-year-reign of human world champion

Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions.

• Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply.

• Othello: human champions refuse to compete against computers, who are too good.

• Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.

106

Summary

• Games are fun to work on!

• They illustrate several important points about AI

• perfection is unattainable must approximate

• good idea to think about what to think about

107

Artificial Intelligence

Lesson 7

108

Planning

• Traditional search methods does not fit to a

large, real world problem: it’s needed to

define specific states, and not in general.

• We want to use general knowledge

• We need general heuristic

• Problem decomposition

109

STRIPS Algorithm

• Strips – Stands for STanford Research

Institute Problem Solver (1971).

• Strips idea: start from the goal to the start

state

• See example (pdf).

110

STRIPS – Representation

• States and goal – sentences in FOL.

• Operators – are combined of 3 parts: – Operator name

– Preconditions – a sentence describing the conditions that must occur so that the operator can be executed.

– Effect – a sentence describing how the world has change as a result of executing the operator. Has 2 parts:

• Add-list

• Delete-list

– Optionally, a set of (simple) variable constraints

111

Example – Blocks world

Basic operations – stack(X,Y): put block X on block Y

– unstack(X,Y): remove block X from block Y

– pickup(X): pickup block X

– putdown(X): put block X on the table

A

B

C

TABLE

112

Example – Blocks world (Cont.)

operator(stack(X,Y),

Precond [holding(X),clear(Y)],

Add [handempty,on(X,Y),clear(X)],

Delete [holding(X),clear(Y)],

Constr [X\==Y,Y\==table,X\==table]).

operator(pickup(X),

[ontable(X), clear(X), handempty],

[holding(X)],

[ontable(X),clear(X),handempty],

[X\==table]).

operator(unstack(X,Y),

[on(X,Y), clear(X), handempty],

[holding(X),clear(Y)],

[handempty,clear(X),on(X,Y)],

[X\==Y,Y\==table,X\==table]).

operator(putdown(X),

[holding(X)],

[ontable(X),handempty,clear(X)],

[holding(X)],

[X\==table]).

113

STRIPS Pseudo code

STRIPS(stateList start, stateList goals)

1. Set state = start

2. Set plan = []

3. Set stack = goals

4. while stack is not empty do

1. STRIPS-Step()

5. Return plan

114

STRIPS Pseudo code – Cont.

STRIPS-Step()

switch top of stack t :

1. case t is a goal that matches state:

1. pop stack

2. case t is an unsatisfied conjunctive-goal:

1. select an ordering for the sub-goals

2. push the sub-goals into stack

115

STRIPS Pseudo code – Cont.

3. case t is a simple unsatisfied goal

1. choose an operator op whose add-list matches t

2. replace the t with op

3. push preconditions of op to stack

4. case t is an operator

1. pop stack

2. state = state + t.add-list - t.delete-list

3. plan = [plan | t]

116

Versions and Decision points • 3 decision points

– How to order sub-goals?

– Which operator to choose?

– Which object to place in a variable?

• Different versions – Backtracking? (at each decision point)

– Lifted: remain a variable in the stack with no value Vs.

– Grounded: for each variable, a value is assigned

• The original STRIPS – Backtrack only on the order of sub-goals

– Lifted

117

Artificial Intelligence

Lesson 8

118

Outline

• Inductive learning

• Decision tree learning

119

Learning

• Learning is essential for unknown environments,

– i.e., when designer lacks omniscience

• Learning is useful as a system construction

method,

– i.e., expose the agent to reality rather than trying to

write it down

• Learning modifies the agent's decision

mechanisms to improve performance

Learning Paradigms

• Supervised Learning: with a “supervisor”.

Inputs and their supplied outputs by the

“supervisor”

• Reinforced Learning: with “reward” for a

good action, or “penalty” for a bad action.

Self learning.

• Unsupervised Learning: Try to learn, but

it’s unknown if the learning is correct or

not.

120

121

Inductive learning • Simplest form: learn a function from examples

• f is the target function,

An example is a pair (x, f(x))

• Problem: find a hypothesis h

such that h ≈ f

given a training set of examples

• This is a highly simplified model of real learning:

– Ignores prior knowledge

– Assumes examples are given

122

Inductive learning method

• Construct/adjust h to agree with f on training set

• (h is consistent if it agrees with f on all examples)

• E.g., curve fitting:

123

Inductive learning method

• Construct/adjust h to agree with f on training set

• (h is consistent if it agrees with f on all examples)

• E.g., curve fitting:

124

• Construct/adjust h to agree with f on training set

• (h is consistent if it agrees with f on all examples)

• E.g., curve fitting:

• Ockham’s razor: prefer the simplest hypothesis consistent with data

• The tradeoff between the expressiveness of a hypothesis space and the complexity of finding simple and consistent hypothesis

Inductive learning method

125

Learning decision trees

Problem: decide whether to wait for a table at a restaurant, based on the following attributes:

1. Alternate: is there an alternative restaurant nearby?

2. Bar: is there a comfortable bar area to wait in?

3. Fri/Sat: is today Friday or Saturday?

4. Hungry: are we hungry?

5. Patrons: number of people in the restaurant (None, Some, Full)

6. Price: price range ($, $$, $$$)

7. Raining: is it raining outside?

8. Reservation: have we made a reservation?

9. Type: kind of restaurant (French, Italian, Thai, Burger)

10. WaitEstimate: estimated waiting time (0-10, 10-30, 30-60, >60)

126

Attribute-based representations • Examples described by attribute values (Boolean, discrete, continuous)

• E.g., situations where I will/won't wait for a table:

• Classification of examples is positive (T) or negative (F)

127

Decision trees • One possible representation for hypotheses

• E.g., here is the “true” tree for deciding whether to wait:

128

Expressiveness • Decision trees can express any function of the input attributes.

• E.g., for Boolean functions, truth table row → path to leaf:

• Trivially, there is a consistent decision tree for any training set with one path to leaf for each example (unless f nondeterministic in x) but it probably won't generalize to new examples

• Prefer to find more compact decision trees

129

Decision tree learning • Aim: find a small tree consistent with the training examples

• Idea: (recursively) choose "most significant" attribute as root of

(sub)tree

130

Choosing an attribute

• Idea: a good attribute splits the examples into subsets that

are (ideally) "all positive" or "all negative"

• Patrons? is a better choice

131

Using information theory

• To implement Choose-Attribute in the DTL

algorithm

• Information Content of an answer (Entropy):

I(P(v1), … , P(vn)) = Σi=1 -P(vi) log2 P(vi)

• For a training set containing p positive examples

and n negative examples:

np

n

np

n

np

p

np

p

np

n

np

pI

22 loglog),(

132

Information gain

• A chosen attribute A divides the training set E into subsets E1, … , Ev according to their values for A, where A has v

distinct values.

• Information Gain (IG) or reduction in entropy from the

attribute test:

• Choose the attribute with the largest IG

v

i ii

i

ii

iii

np

n

np

pI

np

npAremainder

1

),()(

)(),()( Aremaindernp

n

np

pIAIG

133

Information gain

For the training set, p = n = 6, I(6/12, 6/12) = 1 bit

Consider the attributes Patrons and Type (and others too):

Patrons has the highest IG of all attributes and so is chosen by the DTL

algorithm as the root

bits 0)]4

2,

4

2(

12

4)

4

2,

4

2(

12

4)

2

1,

2

1(

12

2)

2

1,

2

1(

12

2[1)(

bits 0541.)]6

4,

6

2(

12

6)0,1(

12

4)1,0(

12

2[1)(

IIIITypeIG

IIIPatronsIG

134

Example contd.

• Decision tree learned from the 12 examples:

• Substantially simpler than “true” tree---a more complex

hypothesis isn’t justified by small amount of data

135

Performance measurement • How do we know that h ≈ f ?

1. Use theorems of computational/statistical learning theory

2. Try h on a new test set of examples

(use same distribution over example space as training set)

Learning curve = % correct on test set as a function of training set size.

A learning curve for the

decision tree algorithm on

100 randomly generated

examples in the restaurant

domain. The graph

summarizes 20 trials.

136

Summary

• Learning needed for unknown environments, lazy designers

• Learning agent = performance element + learning element

• For supervised learning, the aim is to find a simple hypothesis approximately consistent with training examples

• Decision tree learning using information gain

• Learning performance = prediction accuracy measured on test set

137

Fresh our memory with PROBABILITY

Lesson 9

138

• Unconditional or prior probability that a

proposition A is true: P(A)

– In the absence of any other information, the probability

to event A is P(A).

– Probability of application accepted:

P(application-accept) = 0.2

• Propositions include random variables X

– Each random variable X has domain of values:

{red, blue, …green}

– P(X=Red) means the probability of X to be Red

Unconditional Probability

139

• If application-accept is binary random variable ->

values = {true,false}

– P(application-accept) same as P(app-accept = True)

– P(~app-accept) same as P(app-accept = False)

• If Status-of-application domain:

{reject, accept, wait-list}

– We are allowed to make statements such as:

P(status-of-application = reject) = 0.2

P(status-of-application = accept) = 0.3

P(status-of-application = wait-list) = 0.5

Unconditional Probability

140

Conditional Probability

• What if agent has some evidence?

– E.g. agent has a friend who has applied with a much weaker

qualification, and that application was accepted?

• Posterior or conditional probability

P(A|B) probability of A given all we know is B

– P(X=accept|Weaker application was accepted)

– If we know B and also know C, then P(A| B C)

141

– P(A B) = P(A|B)*P(B)

– P(A B) = P(B|A)*P(A)

– P(A|B) = P(A B) / P(B)

– P(B|A) = P(A B) / P(A)

B A

Product rule

142

• Probability of all the possible values of X Denote by

P(X)

– Note that P is in bold

– In our example:

X = Status-of-application

Xi {reject, accept, wait-list}

P(X) = <0.2, 0.3, 0.5>

• P(X=xi) = 1

Probability Distribution

143

Joint Probability Distribution

• Joint probability distribution is a table

– Assigns probabilities to all possible assignment of values for combinations of variables

• P(X1,X2,..Xn) assigns probabilities to all possible assignment of values to variables X1, X2,..Xn

144

Joint Probability Distribution

• X1 = Status of your application

• X2 = Status of your friend’s application

• Then P(X1,X2)

0.15 0.3 0.02

0.3 0.02 0.09

0.02 0.09 0.01

X1

X2

Accept Reject Wait-list

Accept

Reject

Wait-list

145

Bayes’ Rule

• Given that

– P(A B) = P(A|B)*P(B)

– P(A B) = P(B|A)*P(A)

P(B|A) = P(A|B)*P(B)

P(A)

• Determine P(B|A) given P(A|B), P(B) and P(A)

• Generalize to some background evidence e

P ( Y | X, e) = P(X | Y, e) * P(Y | e)

P(X | e)

146

Bayes’ Rule Example • S: Proposition that patient has stiff neck

• M: Proposition that patient has meningitis

• Meningitis causes stiff-neck, 50% of the time

• Given:

– P(S | M) = 0.5

– P(M) = 1/50,000

– P(S) = 1/20

– P(M|S) = P(S| M) * P(M) / P(S) = 0.0002

• If a patient complains about stiff-neck,

P(meningitis) only 0.0002

147

Bayes’ Rule

• How can it help us?

– P(A|B) may be causal knowledge, P(B|A) diagnostic knowledge

– E.g., A is symptom, B is disease

• Diagnostic knowledge may vary:

– Robustness by allowing P(B | A) to be computed from others

148

Bayes’ Rule Use

• P(S | M) is causal knowledge, does not change

– It is “model based”

– It reflects the way meningitis works

• P(M | S) is diagnostic; tells us likelihood of M given

symptom S

– Diagnostic knowledge may change with circumstance, thus helpful

to derive it

– If there is an epidemic, probability of Meningitis goes up; rather

than again observing P(M | S), we can compute it

149

Computing the denominator: P(S)

We wish to avoid computing the denominator in the

Bayes’ rule

– May be hard to obtain

– Introduce 2 different techniques to compute (or avoid

computing P(S))

150

Computing the denominator:

#1 approach - compute relative likelihoods:

• If M (meningitis) and W(whiplash) are two possible

explanations:

– P(M|S) = P(S| M) * P(M) / P(S)

– P(W|S) = P(S| W) * P(W)/ P(S)

• P(M|S)/P(W|S) = P(S|M) * P(M) / P(S| W) * P(W)

• Disadvantages:

– Not always enough

– Possibility of many explanations

151

#2 approach - Using M & ~M:

• Checking the probability of M, ~M when S

– P(M|S) = P(S| M) * P(M) / P(S)

– P(~M|S) = P(S| ~M) * P(~M)/ P(S)

• P(M|S) + P(~M | S) = 1 (must sum to 1)

– [P(S|M)*P(M)/ P(S) ] +

[P(S|~M) * P(~M)/P(S)] = 1

– P(S|M) * P(M) + P(S|~M) * P(~M) = P(S)

• Calculate P(S) in this way…

Computing the denominator:

152

The #2 approach is actually - normalization:

• 1/P(S) is a normalization constant

– Must ensure that the computed probability values sum to 1

– For instance: P(M|S)+P(~M|S) must sum to 1

• Compute:

– (a) P(S|~M) * P(~M)

– (b) P(S | M) * P (M)

– (a) and (b) are numerators, and give us “un-normalized

values”

– We could compute those values and then scale them so that

they sum to 1

Computing the denominator

153

Simple Example

• Suppose two identical boxes

• Box1:

– colored red from inside

– has 1/3 black balls, 2/3 red balls

• Box2:

– colored black from inside

– has 1/3 red balls, 2/3 black balls

• We select one Box at random; cant tell how it is colored

inside.

• What is the probability that Box is red inside?

154

Applying Bayes’ Rule

What if we were to select a ball at random from Box, and it is red,

Does that change the probability?

P(Red-box | Red-ball) = P(Red-ball | Red-box) * P(Red-box)

P(Red-ball)

= 2/3 * 0.5 / P(Red-ball)

How to calculate P(Red-ball)?

P(Black-box|Red-ball) = P(Red-ball |Black-box)*P(Black-box)

P(Red-ball)

= 1/3 * 0.5 / P(Red-ball)

Thus, by our approach #2: 2/3 * 0.5 / P(Red-ball) +

1/3 * 0.5 / P(Red-ball)

= 1

Thus, P(Red-ball) = 0.5, and P(Red-box | Red-ball) = 2/3

155

Absolute and Conditional Independence

• Absolute: P(X|Y) = P(X) or P(X Y) = P(X)P(Y)

• Conditional: P(A B | C) = P(A | C) P(B | C)

• P(A| B C)

– If A and B are conditionally independent given C Then, probability of A is not dependent on B

– P(A| B C) = P(A| C)

• E.g. Two independent sensors S1 and S2 and a jammer J1

– P(Si) = Probability Si can read without jamming

– P(S1 | J1 S2) = P(S1 | J1)

156

Combining Evidence • Example:

– S: Proposition that patient has stiff neck

– H: Proposition that patient has severe headache

– M: Proposition that patient has meningitis

– Meningitis causes stiff-neck, 50% of the time

– Meningitis causes head-ache, 70% of the time

• probability of Meningitis should go up, if both symptoms

reported

• How to combine such symptoms?

157

Combining Evidence

• P(C| A B) = P(C A B) / P ( A B)

• Numerator: – P(C A B) = P(B | A C) * P(A C)

= P(B | C) * P(A C)

= P(B | C) * P(A | C) * P (C)

• Going back to our example:

P(M | S H) = P(S| M) * P(H| M) * P(M)

P( S H)

158

Artificial Intelligence

Lesson 10

(From Russell & Norvig)

159

Introduction

• Why ANN

Try to imitate the computational abilities of the human brain.

Some tasks can be done easily (effortlessly) by humans but are hard by

conventional paradigms on Von Neumann machine with algorithmic

approach

• Pattern recognition (e.g, recognition of old friends or simply a hand-

written character)

• Content addressable recall (ASSOCIATIVE MEMORIES)

• Approximate, common sense reasoning (e.g., driving in busy streets,

deciding what to do when we miss the bus)

These tasks are often ill-defined, experience based, hard to apply logic

160

Introduction

Von Neumann machine -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- • One or a few high speed (ns)

processors with considerable

computing power

• One or a few shared high speed

buses for communication

• Sequential memory access by

address

• Problem-solving knowledge is

separated from the computing

component

• Hard to be adaptive

Human Brain -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- • Large # (1011) of low speed

processors (ms) with limited

computing power

• Large # (1015) of low speed

connections

• Content addressable recall

(CAM)

• Problem-solving knowledge

resides in the connectivity of

neurons

• Adaptation by changing the

connectivity

161

• Biological neural activity

– Each neuron has a body, an axon, and many dendrites

• Can be in one of the two states: firing and rest.

• Neuron fires if the total incoming stimulus exceeds the threshold

– Synapse: thin gap between axon of one neuron and dendrite

of another.

• Signal exchange

• Synaptic strength/efficiency

162

Introduction

• What is an (artificial) neural network

– A set of nodes (units, neurons, processing elements)

• Each node has input and output

• Each node performs a simple computation by its node

function

– Weighted connections between nodes

• Connectivity gives the structure/architecture of the net

• What can be computed by a NN is primarily determined

by the connections and their weights

– A very much simplified version of networks of

neurons in animal nerve systems

163

ANN Neuron Models

General neuron model

Weighted input summation

• Each node has one or more

inputs from other nodes, and

one output to other nodes

• Input/output values can be

– Binary {0, 1}

– Bipolar {-1, 1}

– Continuous

• All inputs to one node come in

at the same time and remain

activated until the output is

produced

• Weights associated with links

•

popularmost is

function node theis )(

1

n

i iixwnet

netf

164

•

•

• Step (threshold) function

where c is called the threshold

• Ramp function

Node Function

Step function

Ramp function

.)( :functionIdentity netnetf

.)( :functionConstant cnetf

165

Node Function

• Sigmoid function

– S-shaped

– Continuous and everywhere

differentiable

– Rotationally symmetric about

some point (net = c)

– Asymptotically approach

saturation points

– Examples:

Sigmoid function

When y = 0 and z = 0:

a = 0, b = 1, c = 0.

Larger x gives steeper curve

Perceptron

• The purpose: examples classification:

• Perceptron with N inputs lines gets an

example (x1,…, xn) as input, where each xi is

an attribute value.

• Result=f (x1,…, xn)

• If the result>threshold, return 1, otherwise 0.

• Note: perceptron works only for functions that

are linear separated…

166

167

Perceptrons

• A simple perceptron

– Structure:

• Single output node with threshold function

• n input nodes with weights wi, {i = 1 to n}

– To classify input patterns into one of the two classes

(depending on whether output = 0 or 1)

– Example: input patterns: (x1, x2)

• Two groups of input patterns

(0, 0) (0, 1) (1, 0) (-1, -1);

(2.1, 0) (0, -2.5) (1.6, -1.6)

• Can be separated by a line on the (x1, x2) plane x1 - x2 = 2

• Classification by a perceptron with

w1 = 1, w2 = -1, threshold = 2

168

Perceptrons

• The step function is: • 1, if x>2

• F(x)= { • 0, if x<2

• Implement threshold by a node x0

– Constant output 1

– Weight w0 = - threshold

– A common practice in NN design

(1.6, -1.6) (-1, -1)

169

Perceptrons

• Linear separability

– A set of (2D) patterns (x1, x2) of two classes is linearly

separable if there exists a line on the (x1, x2) plane

• w0 + w1 x1 + w2 x2 = 0

• Separates all patterns of one class from the other class

– A perceptron can be built with

• 3 input x0 = 1, x1, x2 with weights w0, w1, w2

– n dimensional patterns (x1,…, xn)

• Hyperplane w0 + w1 x1 + w2 x2 +…+ wn xn = 0 dividing the

space into two regions

– Can we get the weights from a set of sample patterns?

• If the problem is linearly separable, then YES (by

perceptron learning)

170

• Examples of linearly separable classes

- Logical AND function

patterns (bipolar) decision boundary

x1 x2 output w1 = 1 -1 -1 -1 w2 = 1 -1 1 -1 w0 = -1 1 -1 -1 1 1 1 -1 + x1 + x2 = 0

- Logical OR function

patterns (bipolar) decision boundary

x1 x2 output w1 = 1 -1 -1 -1 w2 = 1 -1 1 1 w0 = 1 1 -1 1 1 1 1 1 + x1 + x2 = 0

x

o o

o

x: class I (output = 1) o: class II (output = -1)

x

x o

x

x: class I (output = 1) o: class II (output = -1)

171

Perceptron Learning Algorithm

1. Initialize weights and threshold:

Set wi(t), (0 <= i <= n), to be the weight i at time t, and ø to be the threshold

value in the output node.

Set w0 to be -ø, the bias, and x0 to be always 1.

Set wi(0) to small random values, thus initializing the weights and threshold.

2. Present input and desired output

Present input x0, x1, x2, ..., xn and desired output d(t). (x0 is always1).

3. Calculate the actual output:

y(t) = fh[w0(t)x0(t) + w1(t)x1(t) + .... + wn(t)xn(t)]

4. Adapts weights

wi(t+1) = wi(t) + η[d(t) - y(t)]xi(t) , where 0 <= η <= 1 is a positive gain

function that controls the adaption rate.

• Steps 3 and 4. are repeated until the iteration error is less than a user-specified

error threshold or a predetermined number of iterations have been completed.

172

• Note:

– It is a supervised learning

– Learning occurs only when a sample input misclassified

(error driven)

• Termination criteria: learning stops when all samples are

correctly classified – Assuming the problem is linearly separable

– Assuming the learning rate (η) is sufficiently small

Perceptron Learning

173

Choice of learning rate: – If η is too large:

– existing weights are overtaken by η[d(t) - y(t)]

– If η is too small (≈ 0): very slow to converge

– Common choice: η = 0.1

• Non-numeric input: – Different encoding schema

ex. Color = (red, blue, green, yellow). (0, 0, 1, 0) encodes

“green”

Perceptron Learning

174

• MLP: Feedforward Networks

– A connection is allowed from a node in layer i only to

nodes in layer i + 1.

– Most widely used architecture.

Conceptually, nodes

at higher levels

successively

abstract features

from preceding

layers

Network Architecture

175

– Generalization: can a trained perceptron correctly classify

patterns not included in the training samples?

• Common problem for many NN learning models

– Depends on the quality of training samples selected.

– Also to some extent depends on the learning rate and

initial weights

– How can we know the learning is ok?

• Reserve a few samples for testing

Perceptron Learning Quality

176

• Examples of linearly inseparable classes

- Logical XOR (exclusive OR) function

patterns (bipolar) decision boundary

x1 x2 output -1 -1 -1 -1 1 1 1 -1 1 1 1 -1

No line can separate these two classes, as can be seen from the fact that the following linear inequality system has no solution

because we have w0 < 0 from

(1) + (4), and w0 >= 0 from

(2) + (3), which is a

contradiction

o

x o

x

x: class I (output = 1) o: class II (output = -1)

(4)

(3)

(2)

(1)

0 0 0 0

210

210

210

210

wwwwwwwwwwww

Linear Separability Again

177

– XOR can be solved by a more

complex network with hidden

units

Threshold 1

Y

z2

z1 x1

x2

2

2

2

2

-2

-2

(-1, -1) (-1, -1) -1 (-1, 1) (-1, 1) 1 (1, -1) (1, -1) 1 (1, 1) (-1, -1) -1

Threshold 0

MultiLayer NN

– Perceptron extension:

1. Hidden layer in addition to input and output layers

2. In the output layer, it’s possible to have more than

one node, e.g., characters classification

3. Activation function: Sigmoid functions and not a

regular step function

4. The functions can be different in each node, but, in

general, use the same function for all the nodes

5. In the input layer, it’s possible to use step function

178

MultiLayer NN-Purpose • Examples classification: possible to classify

more than 2 groups

• Function proximity: f: RnRm.

Input layer with n nodes; output layer with m nodes

• MLP has much higher computational power than

a simple perceptron

• Possible to handle also function that are not

linear separable.

179

180

Multilayer Network Learning Algorithm

181

Backpropagation example

x5

x4

x3 x1

x2

w35

w45

w13

w24

w14

w23

Sigmoid as activation function with x=3:

• g(in) = 1/(1+℮-3·in)

• g’(in) = 3g(in)(1-g(in))

182

Adding the threshold

x5

x4

x3 x1

x0 x6

x2

w35

w45

w13

w24

w14

w23

1

w03

w04

1

w65

183

Training Set

• Logical XOR (exclusive OR) function

x1 x2 output 0 0 0 0 1 1 1 0 1 1 1 0

• Choose random weights

• <w03,w04,w13,w14,w23,w24,w65,w35,w45> = <0.03,0.04,0.13,0.14,-0.23,-0.24,0.65,0.35,0.45>

• Learning rate: 0.1 for the hidden layers, 0.3 for the output layer

184

First Example • Compute the outputs

• a0 = 1 , a1= 0 , a2 = 0

• a3 = g(1*0.03 + 0*0.13 + 0*-0.23) = 0.522

• a4 = g(1*0.04 + 0*0.14 + 0*-0.24) = 0.530

• a6 = 1, a5 = g(0.65*1 + 0.35*0.522 + 0.45*0.530) = 0.961

• Calculate ∆5 = 3*g(1.0712)*(1-g(1.0712))*(0-0.961) = -0.108

• Calculate ∆6, ∆3, ∆4

• ∆6 = 3*g(1)*(1-g(1))*(0.65*-0.108) = -0.010

• ∆3 = 3*g(0.03)*(1-g(0.03))*(0.35*-0.108) = -0.028

• ∆4 = 3*g(0.04)*(1-g(0.04))*(0.45*-0.108) = -0.036

• Update weights for the output layer

• w65 = 0.65 + 0.3*1*-0.108 = 0.618

• w35 = 0.35 + 0.3*0.522*-0.108 = 0.333

• w45 = 0.45 + 0.3*0.530*-0.108 = 0.433

185

First Example (cont) • Calculate ∆0, ∆1, ∆2

• ∆0 = 3*g(1)*(1-g(1))*(0.03*-0.028 + 0.04*-0.036) = -0.001

• ∆1 = 3*g(0)*(1-g(0))*(0.13*-0.028 + 0.14*-0.036) = -0.006

• ∆2 = 3*g(0)*(1-g(0))*(-0.23*-0.028 + -0.24*-0.036) = 0.011

• Update weights for the hidden layer

• w03 = 0.03 + 0.1*1*-0.028 = 0.027

• w04 = 0.04 + 0.1*1*-0.036 = 0.036

• w13 = 0.13 + 0.1*0*-0.028 = 0.13

• w14 = 0.14 + 0.1*0*-0.036 = 0.14

• w23 = -0.23 + 0.1*0*-0.028 = -0.23

• w24 = -0.24 + 0.1*0*-0.036 = -0.24

186

Second Example • Compute the outputs

• a0 = 1, a1= 0 , a2 = 1

• a3 = g(1*0.027 + 0*0.13 + 1*-0.23) = 0.352

• a4 = g(1*0.036 + 0*0.14 + 1*-0.24) = 0.352

• a6 = 1, a5 = g(0.618*1 + 0.333*0.352 + 0.433*0.352) = 0.935

• Calculate ∆5 = 3*g(0.888)*(1-g(0.888))*(1-0.935) = 0.012

• Calculate ∆6, ∆3, ∆4

• ∆6 = 3*g(1)*(1-g(1))*(0.618*0.012) = 0.001

• ∆3 = 3*g(-0.203)*(1-g(-0.203))*(0.333*0.012) = 0.003

• ∆4 = 3*g(-0.204)*(1-g(-0.204))*(0.433*0.012) = 0.004

• Update weights for the output layer

• w65 = 0.618 + 0.3*1*0.012 = 0.623

• w35 = 0.333 + 0.3*0.352*0.012 = 0.334

• w45 = 0.433 + 0.3*0.352*0.012 = 0.434

187

Second Example (cont) • Calculate ∆0, ∆1, ∆2

• Skipped, we do not use them

• Update weights for the hidden layer

• w03 = 0.027 + 0.1*1*0.003 = 0.027

• w04 = 0.036 + 0.1*1*0.004 = 0.036

• w13 = 0.13 + 0.1*0*0.003 = 0.13

• w14 = 0.14 + 0.1*0*0.004 = 0.14

• w23 = -0.23 + 0.1*1*0.003 = -0.23

• w24 = -0.24 + 0.1*1*0.004 = -0.24

188

Summary

• Single layer nets have limited representation power

(linear separability problem)

• Error driven seems a good way to train a net

• Multi-layer nets (or nets with non-linear hidden

units) may overcome linear inseparability problem

189

Artificial Intelligence

Lesson 11

(From Russell & Norvig)

190

Conditional probability

• Conditional or posterior probabilities e.g., P(cavity | toothache) = 0.8

i.e., given that toothache is all I know

• Notation for conditional distributions: P(Cavity | Toothache) = 2-element vector of 2-element vectors)

• If we know more, e.g., cavity is also given, then we have P(cavity | toothache,cavity) = 1

• New evidence may be irrelevant, allowing simplification, e.g., P(cavity | toothache, sunny) = P(cavity | toothache) = 0.8

• This kind of inference, sanctioned by domain knowledge, is crucial

191

Inference by enumeration

• Start with the joint probability distribution:

• Can also compute conditional probabilities:

P(cavity | toothache) = P(cavity toothache)

P(toothache)

= 4.0

0.0640.0160.0120.108

0.0640.016

192

Independence • A and B are independent iff

P(A|B) = P(A) or P(B|A) = P(B) or P(A, B) = P(A) P(B)

P(Toothache, Catch, Cavity, Weather)

= P(Toothache, Catch, Cavity) P(Weather)

• Absolute independence powerful but rare

• Dentistry is a large field with hundreds of variables, none of which are independent. What to do?

193

Conditional independence • P(Toothache, Cavity, Catch) has 23 independent entries

• If I have a cavity, the probability that the probe catches in it doesn't depend on whether I have a toothache: (1) P(catch | toothache, cavity) = P(catch | cavity)

• The same independence holds if I haven't got a cavity: (2) P(catch | toothache,cavity) = P(catch | cavity)

• Catch is conditionally independent of Toothache given Cavity: P(Catch | Toothache,Cavity) = P(Catch | Cavity)

• Equivalent statements: P(Toothache | Catch, Cavity) = P(Toothache | Cavity)

P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity)

194

Bayesian networks • A simple, graphical notation for conditional independence

assertions and hence for compact specification of full joint distributions

• It describes how variables interact locally

• Local interactions chain together to give global, indirect

interactions

• Syntax: – a set of nodes, one per variable

– a directed, acyclic graph (link ≈ "directly influences")

– a conditional distribution for each node given its parents: P (Xi | Parents (Xi))- conditional probability table (CPT)

195

Example 1 • Topology of network encodes conditional independence

assertions:

• Weather is independent of the other variables

• Toothache and Catch are conditionally independent given Cavity

• It is usually easy for a domain expert to decide what direct influences exist

Cavity P(C=true |

Cavity)

T .9

F .05

P(Cavity=true) = 0.8

Cavity P(T=true | Cavity)

T .8

F .4

P(W=true) = 0.4

196

Example 2 • N independent coin flips :

• No interactions between variables: absolute independence

• Does every Bayes Net can represent every full joint?

• No. For example, Only distributions whose variables are

absolutely independent can be represented by a Bayes’ net

with no arcs.

P(X1=tree) = 0.5 P(X2=tree) = 0.5 P(Xn=tree) = 0.5

197

Calculation of Joint Probability

P(x1x2…xn) = Pi=1,…,nP(xi|parents(Xi))

full joint distribution table

• How to build the Bayes net?

• Given its parents, each node is conditionally independent of everything except its descendants

• Thus,

• Every BN over a domain implicitly represents some joint distribution over that domain

198

Example 3

• I'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar?

• Variables: Burglary, Earthquake, Alarm, JohnCalls, MaryCalls

• Network topology reflects "causal" knowledge:

– A burglar can set the alarm off

– An earthquake can set the alarm off

– The alarm can cause Mary to call

– The alarm can cause John to call

199

Example contd.

For example, what is the probability that there is a burglary, earthquake, alarm,

Jon call, Mary doesn’t?

P(b,e,a,j,~m)=P(b)*P(e)*P(a|b,e)*P(j|a)*P(⌐ m|a)

200

Answering queries

• I'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar?

– P(b|j,⌐m) = P(b,j,⌐m)/P(j,⌐m) (based on p(a|b)=p(a,b)/p(b))

– P(b,j ⌐m) = P(b,e,a,j,⌐m) + P(b,⌐e,a,j,⌐m) + P(b,e,⌐a,j,⌐m) + P(b,⌐e,⌐a,j,⌐m) =

P(b)P(e)P(a|b,e)P(j|a)P(⌐m|a) +

P(b)P(e)P(⌐a|b,e)P(j|⌐a)P(⌐m|⌐a) +

P(b)P(⌐e)P(a|b, ⌐e)P(j|a)P(⌐m|a) +

P(b)P(⌐e)P(⌐a|b, ⌐e)P(j|⌐a)P(⌐m|⌐a)

– Do the same to calculate P(⌐b,j ⌐m) and normalize

P(b|j,⌐m)+ P(⌐ b|j,⌐m)=1

P(b,j,⌐m)+P(⌐ b,j,⌐m)= P(j,⌐m)

201

Laziness and Ignorance

• The probabilities actually summarize a potentially infinite set of circumstances in which the alarm might fail to go off

– high humidity

– power failure

– dead battery

– cut wires

– a dead mouse stuck inside the bell

• John or Mary might fail to call and report it

– out to lunch

– on vacation

– temporarily deaf

– passing helicopter

202

Compactness • A CPT for Boolean Xi with k Boolean parents has 2k rows for the

combinations of parent values

• Each row requires one number p for Xi = true (the number for Xi = false is just 1-p)

• If each variable has no more than k parents, the complete network requires O(n · 2k) numbers

• I.e., grows linearly with n, vs. O(2n) for the full joint distribution

• For burglary net, 1 + 1 + 4 + 2 + 2 = 10 numbers (vs. 25-1 = 31)

• We utilize the property of locally structured system:

local connections between the variables, so each variable won’t have too many parents.

The number of parents depends of how the net is built.

Worst case: n · 2k is equal to 2n

203

Causality?

• Rain (a) causes Traffic (b)

• Let’s build the joint: p(a,b)=p(a|b)*p(b)=p(b|a)*p(a)

204

Reverse Causality? • Both nets are legal, but the previous one is preferred.

Rain cause traffic in general, tough there is a connection between traffic and rain….

205

Causality?

• What do the arrows really mean?

• Topology may happen to encode causal structure

• Topology really encodes conditional independencies

• When Bayes’ nets reflect the true causal patterns:

– Often simpler (nodes have fewer parents)

– Often easier to think about

– Often easier to elicit from experts

• BNs need not actually be causal

– Sometimes no causal net exists over the domain

– E.g. consider the variables Traffic and RoofDrips

– End up with arrows that reflect correlation, not causation

206

Example 2, Again What if the net is build not in a logical order The net looks much more complicated.

Consider the following 2 orders for insertion:

• (a) MaryCalls, JohnCalls, Alarm, Burglary, Earthquake

– Since, P(Burglary|Alarm, JohnCalls, MaryCalls) = P(BurglarylAlarm)

• (b) Mary Calls, JohnCalls, Earthquake, Burglary, Alarm.

207

Connection Types

X ind. Z, given Y? X ind. Z? Diagram Name

Yes Not necessarily B A M Causal chain

Yes No A

J M

Common Cause

No Yes B E

A

Common Effect

208

Test Question

P(H=true) = 0.1

G H

R

J

H P(G=true | H)

T .4

F .8

H G P(R =true | H, G)

false false 0.2

false true 0.9

true false 0.3

true true 0.8

R P(J=true | R)

false 0.2

true 0.7 H - Hardworking

G - Good Grader

R - Excellent Recommendation

J - Landed a good Job

209

What can be inferred?

i:

ii

iii

Q: What is the value of P(H,G,¬R,¬J)?

A: P(H,G, ¬R, ¬J) = P(H)*P(G|H)*P(¬R|H,G)*P(¬J|H,G,

¬R) = P(H)*P(G|H)*P(¬R|H,G)*P(¬J| ¬R) = 0.1 * 0.4 * 0.2

* 0.8 = 0.0064

Q: What if we want to add another parameter, C= Has The

Right Connections?

,P H G P H P G

,P J R H P J R

P J P J H

210

Answer

P(H=true) = 0.1

G H

R

J

H P(G=true | H)

T .4

F .8

C H G P(R =true | H, G,C)

false false false ??

false false true ??

false true false ??

false true true ??

true false false ??

true false true ??

true true false ??

true true true ??

C

P(C=true) = ???

R P(J=true | R)

false 0.2

true 0.7

211

Reachability (the Bayes Ball) Given bayes net, source node and target node, are these two

nodes independent?

• Shade evidence (things that happened) nodes

• Start at source node

• Try to reach target by search

• States: node, along with previous arc

• Successor function:

– Unobserved nodes:

• To any child of X

• To any parent of X if S is coming from a child

– Observed nodes:

• From parent of X to parent of X

• If you can’t reach a node, it’s conditionally independent of

the start node. If there is a path, they are probably

dependent.

212

Example

• L ind. T’, given T?

Yes

• L ind. B?

Yes

• L ind. B, given T?

No

• L ind. B, given T’?

No

• L ind. B, given T and R?

Yes

213

Naïve Bayes

• Conditional Independence Assumption: features are independent of each other given the class:

• What can we model with naïve bayes?

• Any process where,

• Each cause has lots of “independent” effects

• Easy to estimate the CPT for each effect

• We want to reason about the probability of different causes given observed effects

)|()|()|()|,,( 211 CXPCXPCXPCXXP nn

C

X1 X2 Xn X3 …

214

Naive Bayes Classifiers

Task: Classify a new instance D based on a tuple of attribute values into

one of the classes cj C

According to Rule Bayes

Since the denominator is fix

nxxxD ,,, 21

),,,|(argmax 21 nCc

MAP xxxcPc

),,,(

)()|,,,(argmax

21

21

n

n

Cc xxxP

cPcxxxP

)()|,,,(argmax 21 cPcxxxP nCc

215

Summary

• Bayesian networks provide a natural

representation for (causally induced)

conditional independence

• Topology + CPTs = compact representation

of joint distribution

• Generally easy for domain experts to

construct

216

Artificial Intelligence

Lesson 12

• Many applications:

– Floor cleaning, mowing, de-mining, ….

• Many approaches:

– Off-line (getting a map in advance) or On-line

– Heuristic or Complete (promise complete coverage)

• Multi-robot, motivated by robustness and efficiency

Robotics, a Case Study - Coverage

217

• Feels the environment using sensors.

• Has calculations abilities.

• Knows to execute specific operations.

• Uses all the above: feels the environment,

processes it and then decides what it is

needed to operate and then does it.

A Robot….

• Influence: – Dynamic: The environment is changed even tough the robot did no action.

– Static: If the robot did no action, the environment is not changed.

• Feel:

– Accessible: The robot can feel everything in the environment.

– Inaccessible: The robot can feel only specific factors in the environment. The other

factors remain hidden.

• Expected result:

– Non-Deterministic: The robot action’s result is only one from various results options.

– Deterministic: The expected result of a robot is the expected change.

• Possible values of actions and feelings:

– Discrete: The actions and feelings of the robot are discrete, i.e., it is clearly separated

one from the other, and it’s limited on the number of these actions and feelings.

– Continues: The actions and feelings are continues, i.e., there are unlimited possible

values.

Robots Environment Parameters

• Static: If the robot did no action, the environment is not changed.

– to be able to guarantee completeness

• Inaccessible: The robot can feel only specific factors in the

environment.

– greater impact on the on-line version

• Non-deterministic (move 5M, but able to move 5.1M)

• Continuous: actions and feelings have continues values

– Exact cellular decomposition: exact shapes not necessarily in the

same size

– Approximate cellular decomposition: squares in the same size

Environment Assumptions

220

• The purpose: cover a specific area using a few robots that will build a spanning tree of the area.

• Complete - with approximate cellular decomposition

• Robust

– Coverage completed as long as one robot is alive

– The robustness mechanism is simple

• Off-line and On-line algorithms – Off-line: the map is known in advance. It’s possible to plan and improve.

o Analysis according to initial positions

o Efficiency improvements

– On-line: The map is not known. The robotics need to find the map while running.

o Implemented on simulation of real-robots

MSTC- Multi Robot Spanning Tree

Coverage

221

Off-line Coverage, Basic Assumptions

• Area division – n cells

• k homogenous robots

• Robots movement

222 222

STC: Spanning Tree Coverage (Gabrieli and Rimon 2001)

ואין , Gהמכיל את כל צומתי , Gשל קשיר תת גרףהוא G קשיר גרףעץ פורש של , תורת הגרפיםב•

.עץגרף כזה הוא -תת. לו מעגלים

• Area division

• Graph definition

• Building the spanning tree

223 223

Non-backtracking MSTC

• Initialization phase: Build STC, distribute to robots

• Distributed execution: Each robot follows its section

– Low risk of collisions

Robot A is done!

Robot B is done!

Robot C is done!

A

B

C

224 224

• Coverage completed as long as one robot is alive

• Low communication is needed. No need for the robots re-allocation

Guaranteed Robustness

A

B

C

225

Analysis: Non-backtracking MSTC

• Running time = max i k step(i)

• Best case:

• Worst case: n – k

– Unfortunately, common case

1

k

n

A

D

B

C

A

D B

C

226 226

Backtracking MSTC • Similar initialization phase

• But here:

– robots backtrack to assist others

– No point is covered more than twice

A

D

C

B

A

D

C

B

227 227

Backtracking MSTC (cont.)

• Same robustness mechanism: coverage is promised as

long as one robot is alive.

• Same low communication requirements, no robots re-

allocations.

Robot B is done!

Robot C is done!

Robot A is done!

228 228

Backtracking MSTC Analysis

Best case: The same

Worst case: k=2

k>2

1

k

n

1

3

2n

2

n

A

D

B

C

A

B

229 229

Efficiency in Off-line Coverage

• Off-line: getting a map in advance

• Optimal MSTC- improves the average case

• Heterogeneous robots- flexibility

• Optimal spanning tree- improves the worst case

230

Optimal MSTC

• Similar initialization phase

• Robots backtrack to assist others:

– All the robots can backtrack

– Backtracking on any number of steps

• No point is covered more than twice

• Same robustness mechanism

• Same communication requirements

A

D

C

B

E 231 231

Optimal MSTC (cont.)

• Choose a robot

• Search for the minimum valid solution

– Left search

– Right search

• Complexity:

– Check on all the robots: k

– Each search: O(n logn)

– Validity check: O(k)

– Total: O(k2n logn) A

D

C

B

E

232 232

Heterogeneous Robots

• Different speeds

– Non-backtracking MSTC

– Backtracking MSTC

– Optimal MSTC

• Different fuel/battery time

– Non-backtracking MSTC

– Backtracking MSTC

– Optimal MSTC

233 233

Optimal Spanning tree

• Improves the worst case with all 3 algorithms

• The construction is believed to be NP-Hard

(a) (b)

R3

R1

R2

R1

R2 R3

234 234

Generating a Good Spanning Tree (Believed to be NP-Hard)

A A

B B

C C

A B = 12 cells

B C = 12 cells

C A = 12 cells

A B = 28 cells

B C = 4 cells

C A = 4 cells

235 235

A Heuristic Solution

• Build k subtrees on coarse grid

– Start building subtrees from initial locations

– Add cells to each subtree gradually

– Spread away from other robots (based on Manhattan dist)

• Connect subtrees

– Randomly pick connections between subtrees

– Calculate x in resulting tree

– Repeat k^a times (a is a parameter)

– Report tree yielding minimal x

236

Illustration – Stage 1

Min{3,4} = 3

Min{1,2} = 1

Min{2,3} = 2

237

Example

X =

13 16 17 16

238 238

On-line MSTC

• Same basic assumptions:

– Area decomposition- n cells

– k homogenous robots

– Equal tool size and robot movements

• All the robots know their absolute initial position

• Initialization phase

1. Agreed-upon grid construction

2. Self-localization

3. Locations update

239 239

On-line MSTC (Cont.)

240 240

• Coverage completed as long as one robot is alive

• No need for re-allocation

Guaranteed Robustness

241

• Player/Stage with modeled RV-400 robots

• Localization solutions

– GPS

– Odometry with limited errors

• Agreed-upon grid options

– Big enough work-area

– Dynamic work-area

• Collisions avoidance with bumps

– Random wait

– Communication based

• Limited sensors solution

From Theory to Practice

242

Off-line Algorithms Experiments (1)

• Work area: 30X20 cells, 2400 sub-cells

• Each point represents 100 trials

20

220

420

620

820

1020

1220

1420

1 6 11 16 21 26 31

Number of robots

Co

ve

rag

e t

ime

non-backtracking-random backtracking-random optimal-random best case

243 243

Off-line Algorithms Experiments (2)

• Work area: 30X20 cells with 80 holes, 2080 sub-cells

• Each point represents 100 trials

20

220

420

620

820

1020

1220

1420

1 6 11 16 21 26 31

Number of robots

Co

ve

rag

e t

ime

non-backtracking-random backtracking-random optimal-random best case

244 244

Experimental Results

20

120

220

320

420

520

620

720

820

920

3 13 23

Number of robots

Co

ve

rag

e t

ime

non-backtracking-randombacktracking-random

optimal-random

non-backtracking-Best STCoptimal-Best STC

best case

245

Experimental Results - 27% Obstacles

246

On-line Algorithm Run-time Example

247

On-line Algorithm Experiments

• Random places

• Each point represents 10 trials

00:00:00

00:28:48

00:57:36

01:26:24

01:55:12

02:24:00

02:52:48

03:21:36

03:50:24

04:19:12

2 4 6 8 10

Number of robots

Tim

e

Outdoor environment Indoor environment

248 248

Conclusion

• Complete and robust multi-robot algorithms

• Redundancy vs. efficiency with off-line algorithms

• Optimal MSTC which handle heterogeneous robots

• Implemented on-line MSTC with approximation techniques

249